An error ellipse is an indicatrix

An instrument does not report a point. It reports a position and a covariance — two variances and a correlation, three numbers that describe an ellipse — and the ellipse is drawn on the map as a symbol, or quoted in a legend as a radius, or printed in a report as ±5 m — a name for a quantity rather than a measurement of it.

Carrying that ellipse from the ground onto the page is a calculation this site has already done a thousand times under another name.

The two constructions are one construction #

Tissot’s indicatrix is built by taking a small circle on the ground and asking where the projection puts it. The answer is the image of the unit circle under the linear map from ground displacement to page displacement — call that matrix A, whose two columns are where the page goes for one metre east and for one metre north — and the image is an ellipse whose semi-axes are A’s singular values.

An error ellipse is built by taking a covariance Σ on the ground and asking what covariance it becomes on the page. The answer is A Σ Aᵀ.

Set Σ = σ²I, which is what the instrument is equally good in every direction means, and the page covariance is σ² A Aᵀ, whose eigenvalues are the squares of A’s singular values. Those are the indicatrix’s semi-axes, which are the two quantities that survive a change of coordinates. The error ellipse of an isotropic instrument is Tissot’s indicatrix, scaled by σ.

This is not an analogy or a resemblance. It is the same matrix, decomposed the same way, and the site has been drawing it since its first figure without ever saying what else it describes.

What it costs, measured #

Pushing one stated covariance through eight projections at 55° north gives eight different accuracies for one instrument.

Read as a table:

Every one of those ratios equals the projection’s own indicatrix ratio, computed by a completely separate route through h, k and θ, to between 4 × 10⁻¹⁶ and 3 × 10⁻⁸ — the larger residuals belonging to the conformal projections, where the ratio is two nearly equal numbers divided by one another and the arithmetic has nothing left to be precise about. That agreement is the check the figure asserts rather than the claim it makes. The claim is the first column read as a statement about an instrument: a survey good to five metres in every direction is drawn, on an equal-area cylindrical map at 55°, as though it were three times better east–west than north–south.

And it is the conformal projections that keep the shape, which puts an old word to an unfamiliar use. Conformality is normally sold as angles are right, and its consequence for shapes is described in terms of small countries looking like themselves. Here it is a statement about metrology: a conformal projection is the only kind on which a stated circular accuracy is still circular where it is drawn.

Size is the other half, and it goes the other way #

The shape is what conformality protects. The size is what it abandons.

Reading a map’s accuracy symbols is therefore a choice between two ways of being misled, and the choice is made by the projection rather than by the reader.

A symbol on an equal-area map means the same amount of ground everywhere and a different shape of it. A symbol on a conformal map means the same shape and a different amount. Neither map says which, and both draw the symbol identically. This is the trade-off the whole collection is about, arriving from a direction that has nothing to do with countries: the trade-off is forced for accuracy symbols too, because it is forced for every small figure on the map.

Correlation is the third number, and a different property protects it #

A covariance has three independent components, and the ellipse’s shape uses only two of them. The third is the correlation, which says whether an error to the east tends to come with an error to the north — the thing that makes an ellipse lean.

Put a ground covariance of five metres each way with a correlation of 0.7 through four projections at the same point and read the correlation back off the page:

Mercator preserves both, which is expected. Lambert cylindrical preserves the correlation exactly while nearly doubling the axis ratio, which is not: an equal-area cylindrical projection is a diagonal scaling in the east and north directions, and a diagonal scaling leaves a correlation coefficient alone while changing every ellipse it acts on.

So the two halves of the matrix are protected by two different properties. The shape survives where the map is conformal. The correlation survives where the graticule stays orthogonal on the page, which is true of every normal cylindrical projection whether it is conformal or not, and false of every pseudocylindrical one. Mollweide, which is the better world map by any ordinary criterion, is the worst of the four at both — it loses sixteen per cent of the correlation and a fifth of the axis ratio, in opposite directions.

There is no projection on which all three numbers survive, for the same reason there is no faithful map: preserving the whole of A Σ Aᵀ for every Σ requires A to be a similarity everywhere, which is an isometry up to scale, which the curvature forbids.

The inverse case, which is worse #

If a projection turns a circle into an ellipse, it turns some ellipse into a circle. That ellipse is a real instrument.

A reader measuring that symbol on the page recovers an accuracy that is equally good in every direction, and the instrument’s directional weakness — the thing an error ellipse exists to communicate — has been removed by the map it was drawn on.

The cancelling error is not a contrived one. Its axis ratio must be the projection’s indicatrix ratio and its orientation must be the indicatrix’s principal direction, and both are ordinary values: 3.04 to 1 lying east–west at 55° on a common equal-area map. On Mollweide at the same point the cancelling error is 1.33 to 1 lying 65° from the meridian, which is a shape a real network of observations produces without trying.

The condition for a map to be honest about a stated accuracy is not that its distortion is small. It is that its distortion is isotropic — that its indicatrix is a circle — which is conformality and nothing else. An equal-area map with excellent areal fidelity and modest angular deformation is still lying about the shape of every error ellipse on it, by exactly its angular deformation.

What was computed, and how #

Three quantities and one identity.

The linear map A is localFrame, which returns the page displacement produced by one metre of ground displacement east and one metre north. It is built from the projection’s four partial derivatives divided by the metric’s own M and N, so it is correct on the ellipsoid as well as the sphere without a second code path.

The page covariance is A Σ Aᵀ with A scaled by the Earth’s radius so that Σ can be quoted in metres. Its ellipse is the eigen-decomposition of a 2 × 2 symmetric matrix, done in closed form.

The identity is asserted rather than assumed: for every projection in the ranking figure, the axis ratio of the propagated covariance is compared with a/b from the distortion machinery, computed by an entirely separate route through h, k and θ, and required to agree to a part in a million. The two agree to nine figures on all eight, which is what makes the claim a measurement rather than a rearrangement of symbols.

The assertion also has to refuse. It requires the ratio to be 1 to a part in a million on the conformal projections and greater than 1.001 on the others — so a broken localFrame returning the identity matrix would pass the first half and fail the second.

The cancelling-error figure builds its covariance in the indicatrix’s own principal frame, using the ground azimuth principalDirection reports, rather than in the graticule’s frame. That distinction is invisible on a normal cylindrical projection, where the two frames coincide, and is the whole construction on anything else — a first version aligned to the meridian produced a circle on Lambert cylindrical and an ellipse on Mollweide, which was the arithmetic pointing out that distortion has a direction.

The legend that would be honest #

A map that reports positional accuracy has three options and only one of them is unambiguous.

It can draw a symbol and let the reader measure it, which means what the symbol stands for varies across the sheet by everything measured above. It can print a number in the legend — ±5 m — which is a statement about the ground and is then not what the symbols show. Or it can do what almost nothing does and say which of the two the symbol is: a fixed page size standing for a varying ground accuracy, or a fixed ground accuracy drawn at a varying page size.

The second of those is what a reader assumes and the first is what software produces, because a rendering pipeline draws symbols in page units. That mismatch is not a bug anybody introduced; it is the default behaviour of every drawing system, applied to a quantity that lives on the ground.

The ellipse’s area is the areal factor, exactly #

The essay’s two halves — shape protected by conformality, size not — have a third quantity between them, and it has the cleanest statement of the three.

The page covariance is A Σ Aᵀ, so its determinant is (det A)² det Σ. The area of an error ellipse is proportional to the square root of the determinant of its covariance, so

$$\text{drawn ellipse area} = |\det \mathbf{A}| \times \text{ground ellipse area},$$

and |det A| is the areal scale factor. An error ellipse is drawn at its true ground area exactly when the projection is equal-area, whatever its shape does.

That completes a symmetry the essay’s two readback figures show and do not name:

So the two families are exactly complementary about error symbols, and neither is better. What differs is which reading of a symbol a reader can trust. On an equal-area map, comparing two accuracy symbols by how much ink they occupy is correct — and comparing their shapes is not. On a conformal map the reverse.

The arithmetic is worth having because it makes one common practice defensible and another indefensible. A reader who scans a sheet and forms an impression of how much of this map is well surveyed is integrating area, and on an equal-area map that impression is right — the total ink devoted to uncertainty is proportional to the total ground area whose position is uncertain, at every latitude, exactly. On a conformal map the same glance over-weights the high latitudes by the areal factor, which at 80° is a factor of thirty-three.

A reader who instead looks at one symbol and asks which direction is this position weak in is reading the shape, and that reading is right on the conformal map and wrong on the equal-area one — by the projection’s own angular deformation, which is 60.6° at 55° north on the Lambert cylindrical.

Neither reading is unreasonable and no map supports both, which is the trade-off arriving for the third time in this essay and for the same underlying reason: the shape and the area of a small figure are governed by a/b and by ab, the two invariants cannot both be pinned, and an error ellipse is a small figure like any other.

There is one practical consequence worth extracting. Where accuracy symbols are the point of a map — a coverage plot, a data-quality figure, a survey report — the choice of projection is a choice about which question the reader is allowed to ask, and it should be made deliberately and stated in the caption. An equal-area sheet for a map about how much ground is well covered; a conformal one for a map about which directions are weak. That is the same instruction purpose before property gives everywhere else, arriving at a kind of map nobody thinks of as a thematic one.

Where the model stops #

The propagation is linear, so it inherits the limits of a linear model of the projection at a point. That is the same restriction the indicatrix is a limit established for the distortion ellipse: it is exact in the limit of a vanishing error and approximate for a finite one, with the departure growing as the square of the ellipse’s size relative to the sphere. For a five-metre error the linear model is exact to about one part in 10¹²; for a five-kilometre one it is exact to about one part in a million; for a scatter spread over a continent it is not a description at all, which is the subject of the next rung.

A covariance is a two-parameter description of a distribution that may have more parameters than that. A multipath error is not Gaussian, a network with an undetected blunder is not Gaussian, and no amount of correct propagation makes an ellipse the right summary of either. What propagates correctly here is the second moment, and only that.

And the vertical is absent. A three-dimensional covariance has six independent components, a projection is a map of two of the three coordinates, and the height component travels through a different chain entirely — the third coordinate moves too, and it moves under its own rules.

The generalisation #

Every reported uncertainty is expressed in some coordinate system, and changing the coordinate system changes the uncertainty’s shape by exactly the Jacobian of the change. That is the whole of propagation of error in one sentence, and it is why the same sentence appears in metrology, in statistics as the delta method, and in physics as the transformation of a covariance under a change of frame.

What cartography contributes is a case where the Jacobian is not a small correction. A change of units multiplies a covariance by a number; a change of projection multiplies it by a matrix whose condition number reaches 3 at ordinary latitudes and grows without limit near a projection’s own singularities. The transformation is the dominant term, and it is invisible because the map does not report it.

One further consequence of the determinant identity is worth stating for a producer rather than a reader. Because the drawn area carries |det A| exactly, a rendering pipeline that wants error symbols to be comparable by eye across a world sheet has a one-line repair available on any projection: divide each symbol’s linear size by the square root of the areal factor at its own position. That restores the true ground area everywhere, on a conformal map as well as an equal-area one, at the cost of the symbols no longer sharing a page size — which is the same trade a proportional symbol makes, and it is available here for the same reason and by the same multiplication.

Who found it, and when #

Tissot published the indicatrix in 1881 as a way of describing distortion, and error ellipses entered surveying through the least-squares adjustment tradition that runs from Gauss’s 1809 Theoria motus to Helmert’s work on network adjustment in the 1870s and 1880s. The two lines of thought are contemporaneous, they use the same 2 × 2 matrices, and they were developed by people who read each other — Helmert’s name is on the seven-parameter transformation this site uses constantly.

They were nevertheless kept apart, because they answered questions asked by different professions: one described a map, the other described a measurement. The apparatus for combining them has existed for a century and a half, and the reason to combine them is modern — accuracy is now a field in a data format, carried automatically from a receiver into a rendering nobody chose.

Where the ladder goes next #

Both rungs so far have been about a single position: how wide it is written, and what shape its uncertainty is drawn as. The next asks what happens when several noisy positions are combined on the page — and finds an error that does not shrink when more observations are added, because it is not noise at all.